1. Introduction

Surface plasmon polaritons (SPPs) have the capabilities in overcoming the diffraction limit of light, manipulating light in subwavelength volumes and enhancing light-matter interactions, which give remarkable insights into the miniaturization and integration of photonic devices with desirable functionalities [1–3]. Such plasmonic devices have been observed in various nanostructures such as metal particles [4,5], hybrid systems [6,7], metamaterials [8–10], and metal-insulator-metal (MIM) waveguides [11–13]. Among them, the MIM-based nanodevices are widely adopted due to their unique optical field localization and propagation capabilities [14–16]. Numerous MIM-based nanodevices have been investigated and analyzed both theoretically and experimentally, such as wavelength demultiplexers [17,18], nanosensors [19,20], optical switches [21,22], and filters [23,24]. Moreover, Fano resonance and electromagnetically induced transparency (EIT) like responses can also be achieved in MIM-based systems [25–28]. In recent years, electromagnetically induced absorption (EIA) based on MIM waveguides has gradually attracted more and more attention because of its aberrant dispersion and novel fast-light features, which are of great importance in the research and study of nanophotonics [29–33]. Li et al. proposed two kinds of MIM-based structures to realize EIA [29,30]. Wen et al. simulated an end-coupled resonator and a ring-groove system to study EIA [31,32]. Liu et al. investigated a double-ring resonator system to achieve tunable multimode EIA [33]. Although the combination of EIA and MIM-based SPPs has been both theoretically and experimentally analyzed and been proved to be a potential candidate for realizing on-chip integrated fast light optical devices, the EIA phenomenon in a compact MIM-based structure has only been sparsely explored. Further study of EIA phenomena in simple MIM waveguide systems, as well as multiple EIA, EIT and Fano resonance, will contribute to the functions of fast light, slow light and high sensitivity sensing in different wavelengths, which holds great significance for the realization of complex, multi-functional and high-performance nano-integrated devices.

In this paper, a compact MIM-based waveguide system consisted of a square cavity and a slot resonator is proposed, of which the optical characteristics are analyzed by the finite element method and confirmed by the coupled mode theory. Simulation results show that the system’s EIA features can be easily tuned, and by adding another slot resonator, double EIA valleys can be observed. About ∼ -1 ps optical delay is achieved owing to the anomalous dispersion at the EIA valley. When the coupling distance g = 0 nm, dual Fano resonances are generated as a result of the structure’s symmetry breaking and the excitation of the antisymmetric mode, which yields a plasmonic nanosensor with a sensitivity of about S = 1200 nm/RIU and figure of merit, FOM = 16600. Compared to the previously reported MIM-based EIA works [29–33], our system is simpler and more easily scalable to achieve dual-EIA and EIT, enabling fast light and slow light effects at different wavelengths. Furthermore, our compact system has been expanded to realize dual Fano resonances, which can be used for high-sensitivity refractive index sensing devices. These unique and exotic properties lead to fascinating potential applications, such as ultra-compact plasmonic devices in highly all-optical integration systems, especially for high-performance fast-light on chip devices.

2. Structure and simulation results

As shown in Fig. 1, the investigated structure comprised of a slot cavity (length D and width H) and a square cavity (side-length L) with a coupling distance g in the MIM system. The width of the bus waveguide is fixed at w = 50 nm. Here, the blue and white areas denote the Ag with dielectric constant (ε_m) taken from the literature [34] and air (ε_air= 1.0) materials, respectively. The temporal coupled mode theory (CMT) is applied to analyze the optical properties for a further qualitative understanding of the proposed structure [35,36]. The amplitude of the square cavity (a) and slot resonator (b) are normalized to the energy in the modes. The amplitudes of the incoming S_i, reflecting S_r, and outgoing S_t waves into the cavity are also normalized to the power carried by the waveguide mode. In steady state, the time evolution of the cavity amplitudes can be described as [35,36]

When we remove the slot resonator, the system becomes a band-pass filter structure with

 

Fig. 1. Schematic of the optical system and the geometrical parameter symbols.

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To verify the CMT analysis above, the optical properties of the proposed system are numerically investigated by COMSOL Multiphysics 5.6 based on the finite element method (FEM), and the detailed modeling set can be found in Appendix B. The transmittance (T) of the system is defined as the quotient between the SPP power flows (obtained by integrating the Poynting vector over the channel cross section) of the Port 1 and Port 2 [37], which can be extracted from the COMSOL software built-in S-parameter, see Appendix B. In the simulations, L = 500 nm, D = 300 nm, H = 50 nm, w = 50 nm, and g = 10 nm. The calculated transmission spectra of the system with (red line) and without (black line) the slot resonator are displayed in Fig. 2(a). It is evident that the transmission spectrum exhibits a band-pass filter like Lorentz type line (black line) with a transmitted peak at λ=1100 nm [37]. The simulation results are in good accordance with the theoretical analysis based on CMT. While an absorption valley emerges at the original transmitted peak when the slot resonator above the square cavity is introduced, which could be viewed as a typical EIA response [29–33] because of the constructive interference between two different paths of SPPs. One is that the SPPs directly transmit to ‘Port 2’ through the square cavity from ‘Port 1’, and the other is that the SPPs indirectly transmit to ‘Port 2’ after power exchange between the square cavity and the slot resonator. Accordingly, the reflection spectrum of the system is a typical EIT phenomenon [27,28], and the detailed information can be found in Appendix C. Figure 2(b) shows the phase response of the system, a π phase change can be observed near λ=1086 nm due to the coupling between the square cavity and the slot resonator when the slot resonator is added, which results in a great difference in the transmission spectrum (The original resonant peak becomes an absorption valley). Figure 2(c) provides the field distributions of |Hz|² at λ=1100 nm without the slot resonator while Fig. 2(d) evidences the result at λ=1086 nm with the slot resonator. It is clear that the power in the square cavity is absorbed by the slot resonator with a slight blue shift of the resonance wavelength. The fitting line calculated by CMT based on Eq. (3) is in perfect agreement with the FEM calculation, further details can be found in Appendix C. Moreover, two-singularity points, A (λ=832 nm) and B (λ=1145 nm) in Fig. 2(a), are caused by structural symmetry breaking and the excitation of antisymmetric mode (A) and angle mode (B) [17]. More details about the two-singularity points can be found in Appendix C.

 

Fig. 2. Transmission spectra (a) and corresponding phase response (b) of the system with (red line) and without (black line) the slot resonator. Field maps of |Hz|² at (c) λ=1100 nm without and (d) λ=1086 nm with the slot resonator. Here, L = 500 nm, D = 300 nm, H = 50 nm, and g = 10 nm.

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3. Tuning characteristics of EIA

Additionally, we investigate the influences of the slot resonator on the EIA of the structure based on FEM by changing only one parameter at a time, while keeping the other parameters the same as in the caption of Fig. 2(a). The calculated transmission spectra for various length D, coupling distance g, width H, and refractive index n filled in the slot resonator are shown in Fig. 3(a)-(d), respectively. It can be seen that with increasing D and n, the valley position of EIA is red-shifted, and the resonance wavelength has a linear relationship with D with Δλ/ΔD = 30 nm/10 nm. With increasing g and H, the position of the EIA valley has blue-shifted. In particular, if the coupling distance g is greater than the skin depth of the SPP in silver, the slot resonator will no longer function, i.e., the EIA will disappear. Therefore, by adjusting the parameters of the slot resonator, different EIA valleys can be obtained. It should be noted that we are not talking about the cases where g < 10 nm and H < 50 nm, because If g < 10 nm, the slot resonator is strongly coupled with the square cavity, and the transmission spectrum changes significantly. If the width of the resonator H < 50 nm, the coupling between slot resonator and square cavity becomes very weak. This selection of the range of values of g and H discussed in Fig. 3(b) and 3(c) are consistent with that reported in previous literatures [11,25,29,37].

 

Fig. 3. Transmission spectra for different (a) length D, (b) coupling distance g, (c) width H and (d) refractive index n filled in the slot cavity.

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4. Structure expansion and fast light effect

By adding another slot resonator, we expand the structure successively, which is illustrated in Fig. 4(a) and 4(b). Figures 4(c)-(g) inform the transmission spectra of the system with different D₂ at D₁= 300 nm, when the additional slot resonator is placed on the other side (down side) of the square cavity (Fig. 4(a)). It can be seen that when D₁≠D₂, an extra EIA valley appeared in the system, whose location changes linearly as D₂ increasing (green circles in Figs. 4(c)-(g)), as shown by the black line in the figure; whereas the valley corresponding to the original slot resonator remaining unchanged, as indicated by the gray box. Additionally, the mismatch between the two slot resonators causes the SPPs power flows to reflect back and forth between the two slot resonators, creating a new F-P resonator with strong light waves at the output, which appears as an EIT peak in the middle of the two EIA valleys [38–40]. When D₁= D₂, the structure is symmetrical with only one EIA valley. Figures 4(k)-(o) show the corresponding |H_z|² field distribution maps at the location of the extra EIA valley (green circles in Figs. 4(c)-(g), respectively), it can be seen that the light field power is mainly concentrated in the lower slot resonator (the extra EIA valley locates at λ=942 nm, 1014 nm, 1086 nm, 1158 nm, and 1230 nm for D₂= 250 nm, 275 nm, 300 nm, 325 nm, and 350 nm, respectively). Figures 4(h)-(j) show the calculated transmission spectra when two slot resonators are on the same side (Fig. 4(b)). In each transmission spectrum, there are two EIA valleys, which keep a linear relationship with D₁ and D₂ respectively, present as the blue and green lines in Figs. 4(h)-(j). When the parameters of the upper and lower slot resonators are switched, the transmission spectrum undergoes a mild change, which is the result of the phase reversal of the system in the lower slot resonator, as can be seen from the field maps of H_z in Fig. 4(p)and 4(q) (the EIA valleys are λ=942 nm and 1257 nm for D₁= D₂= 300 nm, denoted by the blue circles). The expansion of the structure enriches the transmission characteristics of the system and enables a wider application of this structure in the field of nano-photonics.

 

Fig. 4. The extended structures: (a) The slot resonator is on the opposite side of the square cavity. (b) The slot resonator is on the same side of the square cavity. (c)-(g) Transmission spectra of Fig. 4(a) for different D₂ ∼ (250 nm, 275 nm, 300 nm, 325 nm, 350 nm) at D₁= 300 nm, respectively. (h) Transmission spectra of Fig. 4(b) for D1 = 280 nm, D2 = 300 nm (black line) and D₁= 300 nm, D₂= 280 nm (red line). (i) Transmission spectrum of Fig. 4(b) for D1 = 300 nm, D₂= 300 nm. (j) Transmission spectra of Fig. 4(b) for D1 = 320 nm, D2 = 300 nm (black line) and D₁= 300 nm, D₂= 320 nm (red line). (k)-(o) Corresponds to the field maps of |H_z|² at the wavelength of the green circle in Fig. 4(c)-(g), respectively. (p)-(q) Field maps of H_z at λ=942 nm, and λ=1257 nm in Fig. 4(i), respectively. Here, L = 500 nm, H = 50 nm, and g = 10 nm.

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It is well known that the EIT-like phenomenon can support slow group velocities [41,42], whereas, EIA can support fast light due to the aberrant dispersion at the EIA valley [32,33]. The optical delay τ_g can be expressed as [41,42]:

 

Fig. 5. Transmission phase shift (black line) and optical delay line (blue line) in different plasmonic systems. Here, Fig. 5(a) shows the situation when there is only one slot resonator with D = 300 nm; Fig. 5(b) shows the situation when two slot resonators are on the same side of the square cavity with D₁= D₂= 300 nm; Fig. 5(c)-(e) shows the situation of two slot cavities at two sides of the square cavity with (D₁, D₂) = (275 nm, 300 nm), (300 nm, 300 nm) and (325 nm, 300 nm), respectively. The other parameters are fixed at L = 500 nm, H = 50 nm, and g = 10 nm.

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5. Sensing characteristic analysis

To facilitate experimental etching, as well as obtaining practical applications in the future [43,44], an improved structure is designed with a slot resonator directly connected to the square cavity (g = 0 nm), which is displayed in the inset in Fig. 6(a). This novel structural design should facilitate its fabrication by focusing ion beam (FIB) in etching process [45,46]. Other parameters are set as L = 500 nm, D = 300 nm, H = 100 nm and w = 50 nm. As shown in Fig. 6(a), the refractive index sensing properties are numerically calculated to verify the validity of applications exploitability of the structure. It can be clearly seen that two Lorentz and two Fano resonances are emerging in the transmission spectra, which are named LR1 (λ=710 nm), FR1 (λ=818 nm), LR2 (λ=1065 nm), and FR2 (λ=1275 nm), respectively [47,48]. For plasmonic micro and nano structures, when the coupling distance varies between zero and non-zero, there will be an abrupt phase change resulting in significant alterations to the transmission spectrum. Because of the strongly trapped resonance, it will increase the sensitivity of the system as a plasmonic refractive index sensor [49]. The sensitivity S (S=Δλ/Δn) of a plasmonic sensor (nm/RIU) is usually defined as the shift in the resonant wavelength per unit variation of the refractive index [49]. Thus, the sensitivities are about S = 700 nm/RIU, 700 nm/RIU, 1000 nm/RIU, and 1200 nm/RIU for LR1, FR1, LR2, and FR2, respectively. To conduce a more detailed evaluation of the performance of the nanosensor, we investigate the figure of merit (FOM), which is calculated as [50,51]

Figure 6(b) shows the FOM at different wavelengths of 1200 and 16600 at FR1 and FR2, respectively (while about FOM = 0 at LR1 and LR2). The results above depict that FR2 holds a more desirable performance in refractive index sensing. Figures 6(c) and 6(d) show the field distributions of H_z for FR1 and FR2, respectively. It is easy to know that the FR1 is generated by the antisymmetric mode excitation caused by the symmetry breaking of the structure [47] whereas the FR2 is generated by the higher order mode being excited as the longitudinal effective length increases [48]. The results above show that the proposed structure may open up a new route in the field of biosensing with various insights and practical applications [12,13,43,44]. Furthermore, this paper also demonstrates that the sensitivity of the sensor can be effectively enhanced by achieving the Fano shape, which is of great significance and potential in future chemical and biological sensing applications [52,53].

 

Fig. 6. (a) Transmission spectra with different refractive indexes. (b) The calculated FOM at different wavelengths. Field maps of Hz at (c) λ=818 nm (FR1) and (d) λ=1275 nm (FR2). Inset shows the extended structure. Here, L = 500 nm, D = 300 nm, H = 100 nm, and g = 0 nm.

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6. Conclusion

In conclusion, we have demonstrated a new class of MIM-based waveguide system for realizing EIA response, which consists of a square cavity and a slot resonator analyzed by FEM, and confirmed by CMT. Simulation results show that the EIA characteristics of this system can be easily tuned, and double EIA valleys can be observed by adding another slot resonator. Furthermore, the aberrant dispersion at the EIA valley is also been studied, and about ∼ -1 ps optical delay is achieved, which has important applications in the on-chip fast-light area. When the coupling distance g = 0 nm, dual Fano resonances are generated by the symmetry breaking of the structure and the excitation of the antisymmetric mode, which yield a plasmonic nanosensor with a sensitivity of S = 1200 nm/RIU and FOM = 16600. These results could be applied to developing ultra-compact plasmonic devices in highly all-optical integration systems, especially for high-performance fast-light on chip devices.

Appendix A

The skin depth of SPPs in metals (Ag) can be calculated by

(7)$${\delta _m} \approx \frac{\lambda }{{2\pi }}\frac{{\sqrt {|real({\varepsilon _m}) + {\varepsilon _{air}}|} }}{{|real({\varepsilon _m})|}},$$

and generally the skin depth is about 20-50 nm for our calculated wavelengths (for example, at λ=1000 nm, the δ_m= 22 nm obtained by the above equation). Therefore, the coupling distance g is usually less than 50 nm, which is also the coupling distance selected in most reported literatures [37,44].

Appendix B

Because the width w of the bus waveguide is 50 nm so that only the fundamental transverse magnetic mode (TM₀) can exist in the structure. Here, the module Wave Optics (Electromagnetic Waves, Frequency Domain) in COMSOL Multiphysics 5.6 with 2D is used to investigate the optical response of the proposed structure. Numeric ports are set on the left (with Port 1 “on”) and right (with Port 2 “off”) sides, and the up and bottom sides are set with the Scattering Boundary Conditions. Finally, two boundary mode analysis are added to analyze the effective index of the two Ports and Frequency domain to calculate the optical response by extracting the built-in S-parameter for transmission phase shift arg(ewfd.S21), transmission spectrum T = abs(ewfd.S21)² and Reflection spectrum R = abs(ewfd.S11)².

Appendix C

Figure 7(a) shows the transmission curve (red) and reflection curve (black) of the structure in Fig. 1. It can be seen that the transmission curve is a typical EIA response, while the reflection curve is a typical EIT response. The two-singularity points, A (λ=832 nm) and B (λ=1145 nm) in Fig. 7(a), are caused by structural symmetry breaking and the excitation of antisymmetric mode (A) and angle mode (B). In order to verify this claim, another slot cavity is added on the other side of the square cavity, and the two slot cavities are distributed symmetrically (as shown in the inset in Fig. 4(a) with D₁= D₂= 300 nm). The calculated transmission and reflection spectra are shown in Fig. 7(b), where the curve becomes smoother. Figure 7(c) exhibits the fitting line calculated by CMT (blue symbol line) based on Eq. (3), which agrees perfectly with the calculation of FEM (red line). Figure 7(d) and (e) show the field distributions of Hz corresponding for points A (λ=832 nm) and B (λ=1145 nm), respectively. It is apparent that the antisymmetric and angle modes in the system are excited due to symmetry breaking.

 

Fig. 7. Transmission and reflection spectra for the structures in (a) Fig. 1 and (b) the inset in Fig. 4(a) with D₁= D₂= 300 nm. (c) Transmission spectra calculated by FEM (red line) and CMT (blue dashed line). Field maps of Hz at (d) λ=832 nm (point A) and (e) λ=1145 nm (point B).

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Funding

Fundamental Research Funds for the Central Universities (buctrc202143); Program of State Key Laboratory of Quantum Optics and Quantum Optics Devices (KF202202); National Natural Science Foundation of China (11974225, 12174037, 12204030, 62175010).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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